Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Two-dimensional area minimizing integral currents are classical minimal surfaces


Author: Sheldon Xu-Dong Chang
Journal: J. Amer. Math. Soc. 1 (1988), 699-778
MSC: Primary 49F20; Secondary 49F10, 49F22, 58E12, 58E15
MathSciNet review: 946554
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Geometric measure theory guarantees the existence of area minimizing integral currents spanning a given boundary or representing a given integral homology class on a compact Riemannian manifold. We study the regularity of such generalized surfaces. We prove that in case the dimension of the area minimizing integral currents is two, then they are classical minimal surfaces. Among the consequences of this regularity result, we know now that any two dimensional integral homology class on a compact Riemannian manifold can be represented by a finite integral linear combination of classical closed minimal surfaces that have only finitely many intersection points.

The result is proved by using the theory of multiple-valued functions developed by F. Almgren in [A]. We extend many important estimates in his paper and extend his construction of center manifolds. We use the branched center manifolds and lowest order term in the multiple-valued functions approximating the area minimizing currents to construct two sequences of branched surfaces near an interior singular point to separate the nearby singularity gradually. The analysis developed in this paper enables us to conclude the generalized surface must coincide with one of the branched surfaces.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 49F20, 49F10, 49F22, 58E12, 58E15

Retrieve articles in all journals with MSC: 49F20, 49F10, 49F22, 58E12, 58E15


Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1988-0946554-0
PII: S 0894-0347(1988)0946554-0
Article copyright: © Copyright 1988 American Mathematical Society